We study vantage-point trees constructed using an independent sample from the uniform distribution on a fixed convex body $K$ in $(\mathbb{R}^d,\|\cdot\|)$, where $\|\cdot\|$ is an arbitrary homogeneous norm on $\mathbb{R}^d$. We prove that a sequence of sets, associated with the left boundary of a vantage-point tree, forms a recurrent Harris chain on the space of convex bodies in $(\mathbb{R}^d,\|\cdot\|)$. The limiting object is a ball polyhedron, that is, an a.s.~finite intersection of closed balls in $(\mathbb{R}^d,\|\cdot\|)$ of possibly different radii. As a consequence, we derive a limit theorem for the length of the leftmost path of a vantage-point tree.

翻译：我们研究基于凸体$K$（在$(\mathbb{R}^d,\|\cdot\|)$空间中，$\|\cdot\|$为$\mathbb{R}^d$上的任意齐次范数）上均匀分布独立样本构建的视点树。我们证明，与视点树左边界相关的一组集合序列，在$(\mathbb{R}^d,\|\cdot\|)$空间的凸体上构成一个递归Harris链。其极限对象为球多面体，即$(\mathbb{R}^d,\|\cdot\|)$空间中（可能不同半径的）闭球之交，且该交集几乎必然有限。作为推论，我们推导了视点树最左路径长度的极限定理。